Cramer C.J. Essentials of Computational Chemistry Theories and Models

494 14 EXCITED ELECTRONIC STATES

option, owing to high spin contamination in the final wave functions. With DFT, however, this problem does not here arise (see Section 8.5), and at the BVWN5/cc-pVDZ and BLYP/ccpVDZ levels of theory, the A triplet is predicted to be lower in energy than the A triplet by 2.3 and 1.6 kcal mol−1, respectively (Worthington and Cramer 1997). This only slightly overestimates the experimental result of 0.9 kcal mol−1 (Gilles, Lineberger, and Ervin 1993).

Although the A triplet is the lowest energy electronic state of fluorovinylidene having triplet spin, the closed-shell singlet is lower in energy still, and is the ground state. Naturally, then, it too is amenable to an SCF description. Note that there can be no variational collapse of the A triplet to the A singlet, not only because the spatial symmetries of the two wave functions belong to different irreducible representations but also because the spin states are different. The predicted SCF energy difference between the 1A and 3A states at the BVWN5/cc-pVDZ and BLYP/cc-pVDZ levels is 30.9 and 31.6 kcal mol−1, respectively, which compares well with an experimental measurement of 30.4 kcal mol−1.

In the case of the triplet of A symmetry, only the difference in spin states prevents variational collapse of the triplet to the singlet, but that is sufficient. Interestingly, DFT does a reasonably good job in predicting the singlet–triplet splitting between these two states, with BVWN5/cc-pVDZ and BLYP/cc-pVDZ both giving values of 33.2 kcal mol−1, compared to an experimental measurement of 31.3 kcal mol−1. If the fluorine is replaced by a t-butyl group, the same theoretical levels predict analogous splittings of 45.7 and 46.8 kcal mol−1, respectively, compared to an experimental measurement (Gunion and Lineberger 1996) of 45.6 kcal mol−1. These good agreements come in spite of the current formal status of DFT, where the Hohenberg–Kohn theorem has only been proven to apply to the lowest-energy state irrespective of spin in each irreducible representation of the molecular point group (see Section 8.2.1).

Many of the same considerations affecting these vinylidene examples arise in comparing the relative energies of the electronic states of phenylnitrene (Figure 14.3). In this system, there are many different theoretical data available to compare to experiment, which itself is available for the lowest two singlet states. Results from SCF calculations at the HF and DFT levels of theory are listed in Table 14.1, as are results from many additional levels that will be discussed at appropriate points later in the chapter.

DFT levels of theory compare about as well with experiment for the splitting between the 3A2 and 11A1 states of phenylnitrene as for the analogous states of the vinylidenes just discussed. Agreement is nearly quantitative at the BLYP level using a triple-ζ basis set. In general in Table 14.1, the energies of the excited states are predicted to be somewhat lower for equivalent levels of theory when triple-ζ basis sets are used in place of those of double- ζ quality. Such behavior is expected, insofar as ground states tend to have more electron density residing in the close-in valence region than do excited states, and thus the ground states are less demanding in terms of basis-set requirements. Moreover, most basis sets are optimized for ground-state atoms and molecules, so to the extent basis-set limitations affect the calculation, they should disproportionately affect excited states.

The DFT values for the 1A2 state derive from the sum method or projection techniques presented in Section 14.4, and discussion of those values is deferred to that point. As for the 21A1 state, although no experimental measurement is available, comparison to other

a Zero-point vibrational energies are not included in the theoretical energies, but ZPVE differences between alternative electronic states are predicted to be small at the few levels where they have been evaluated.

b Determined from sum method.

c Determined from spin projection. d Johnson and Cramer (2001).

e Smith and Cramer (1996).

f Hrovat, Waali, and Borden (1992); Kim, Hamilton, and Schaefer (1992). g Travers et al. (1992); Ellison, G. B., unpublished results.

levels of theory that would be expected to be reasonably accurate suggests that the DFT predictions are substantially too low. The DFT wave function for the 21A1 state is that of Eq. (14.8) and, as discussed above, is found by fortuitous convergence of the SCF equations for this occupation scheme where variational collapse to the 11A1 state of Eq. (14.7) would otherwise be expected. The apparently rather poor accuracy of the energy for the higher state suggests that this orthogonality issue cannot be ignored here, and the SCF procedure must be regarded as unreliable.

As for the HF level, the SCF approach for the closed-shell singlet states is identical to that in the DFT case (in this instance, the two-determinantal nature of the lower energy openshell singlet requires an MCSCF description, so HF values are not reported for this state). However, both of the closed-shell singlets are subject to large non-dynamical correlation effects (as a consequence, in part, of being so close in energy to one another). Since HF theory is much more sensitive to such correlation than DFT, the energies of these two states are predicted to be much too high. This error is in some sense even worse than it appears, because severe spin contamination in the triplet, which exhibits an expectation value for S2 in excess of 2.7, probably causes it too to be poorly represented at the HF level.

Of course, with HF wave functions in hand, it is possible to carry out post-HF calculations to partially correct for electron correlation effects. The poor quality of the HF wave functions, however, militate against any treatment much less sophisticated than coupled-cluster. At the CCSD(T)/cc-pVDZ level, the predicted energy of the lowest closed-shell singlet is in fair agreement with experiment (other data in the table suggest that use of a triple-ζ basis set would improve the CCSD(T) estimate). The energy of the second closed-shell singlet state

looks to be somewhat low, however, again probably reflecting the non-orthogonal nature of the HF reference for this state compared to the lower energy one.

14.2.2CI Singles

For many singly excited states, SCF calculations are not an option under any circumstances. Within the context of using the orbitals of the ground state to describe the excited state, the simplest way to evaluate the energy of the excited state would be to evaluate the Hamiltonian for the determinant formed after promotion of the excited electron. Such an approach is rarely useful, however, and a significant drawback of these singly excited single-configuration wave functions is that, although each one will be orthogonal to the ground state (because of Brillouin’s theorem, see Section 7.3.1), they are unlikely to be orthogonal to one another. However, as long as we limit our consideration to singly excited states, they can be made to be orthogonal to one another with fairly little computational effort, and in the process better descriptions of the states, and presumably better energies, may be determined.

This orthogonalization is the essence of the technique known as CI singles (CIS) because the CI matrix is formed restricting consideration to only the HF reference and all singly excited configurations (Figure 14.5). The matrix is essentially of size M × N where M is the number of occupied orbitals from which excitation is allowed, and N is the number of virtual orbitals into which excitation is considered. If excitation is allowed to occur with a spin-flip of the excited electron (e.g., permitting generation of triplet excited states from singlet ground states or vice versa; see, for example, Sears, Sherrill, and Krylov 2003) then the size increases, although none of the triplet states have matrix elements with any of the singlet states because of their different spins. Orthogonalization of the CIS matrix takes place only in the space(s) of the excited states, since they do not mix with the HF reference. The orthogonalization provides energy eigenvalues each of which has associated with it an eigenvector detailing the weight of every singly excited determinant in the state. That is, the CIS wave function for each excited state is written as

occupied virtual

ia

Figure 14.5 The CIS procedure diagonalizes the CI matrix formed only from the HF reference and all singly excited configurations. The diagonalization provides energy eigenvalues and associated eigenvectors that may be used to characterize individual states as linear combinations of single excitations

length, these energies converge to the band gap for the one-dimensional solid, a property of considerable interest in solid-state chemistry and physics.

Improvement of ab initio CIS wave functions may in principle be accomplished with perturbation theory, but this tends to be very slowly convergent because the existence of a high-energy occupied orbital and a low-energy hole leads to denominators in perturbation theory expressions analogous to Eq. (7.48) that are very near zero, and thus the terms are very large. Head-Gordon et al. (1994) have proposed a more satisfactory approach where the effect of double excitations on the CIS state energy is evaluated; this method is referred to as CIS(D).

CIS technology has a particularly valuable application that is unrelated to an interest in excited states, per se. In systems where the exact orbital occupation of the ground state is not entirely certain, a CIS calculation allows a determination of whether the ground state has actually been found with respect to single excitations. If a CIS eigenvalue is found that is below the HF energy, then the HF reference either is not the ground state, or it may be, but within the accuracy of HF/CIS theory it is clear that a near degeneracy exists with another state or states. In highly symmetric systems, where orbital mixing is restricted by blocking of the Fock matrix, a check of the stability of any final wave function using this approach is a useful precaution.

A variation on the CIS scheme that is empirical in nature but has been demonstrated to offer surprisingly high accuracy in computational practice involves marrying some aspects of DFT with the CIS methodology (DFT-SCI; Grimme 1996). In particular, all HF orbital energies are replaced by KS equivalents, and the Coulomb integrals and diagonal elements are empirically scaled. Once this is done, the usual diagonalization process provides wave functions and state energies that compare nicely with more rigorous theoretical formulations. This technology has not yet seen widespread use.

14.2.3Rydberg States

A particular type of singly excited state merits mention because of its unique characteristics. A so-called Rydberg state is one where the excited electron has an energy very near the level of the continuum, i.e., it is almost detached. Such states may conveniently be thought of as an electron attached to a molecular cation that acts as a central attractor, much as a nucleus acts as a central attractor in an atom. Thus, formal Rydberg orbitals may be of s, p, d, etc. character with the exact chemical nature of the underlying molecular system secondary in importance to its total charge. The Rydberg orbitals are by nature extremely diffuse compared to valence orbitals. As a result, any attempt to describe a Rydberg state requires an AO basis set that either includes diffuse functions on heavy atoms (and possibly H) or is supplemented by additional basis functions specifically tailored to Rydberg character that are not necessarily atom-centered, e.g., they may take the molecular center of mass as their origin (see, for example, Wiberg, de Oliveira, and Trucks 2002). The latter option is more efficient but introduces complications if gradients are desired for geometry optimization.

14.3 General Excited State Methods

Electronic states that cannot be well described as single excitations from the ground state require more general formalisms than those described thus far for the construction of their wave functions. Such formalisms, insofar as they are general, can certainly be used for states that are well characterized as single excitations as well; they simply tend to be somewhat more demanding in terms of computational resources, making methods like CIS economical alternatives when they can be applied. Some of the methods described below have already been discussed in Chapter 7 in the context of improving the description of the ground-state wave function, while others are specific to excited-state applications.

14.3.1Higher Roots in MCSCF and CI Calculations

In the process of determining the expansion coefficients that define an MCSCF wave function along lines similar to Eq. (7.10), CI calculations are carried out in the space of the orbitals that are active in the MCSCF. Excited states that can be generated by electronic excitations within that active space then have a corresponding root in that limited CI window and, if one chooses, one can variationally optimize the orbitals for a root other than the one of lowest energy, i.e., other than the ground state.

In some instances, the root in question is dominated by a single CSF, allowing one to describe the state conveniently by reference to the ground state. In other instances, however, that will not be the case, and simple relationships between the excited state and the ground state cannot be easily formulated. This is, of course, purely a conceptual problem – the wave functions themselves are perfectly well defined and useful.

In any case, once a root is specified, the MCSCF process minimizes the energy for that root following the usual variational procedure. Problems can arise, however, along the way. Consider the situation illustrated in Figure 14.6, with two states having curves that cross along some geometrical coordinate. The point of crossing is a so-called ‘conical intersection’ in the corresponding PESs. In diatomics, such intersections are not permitted if the curves correspond to electronic states of the same symmetry (the ‘non-crossing rule’), but in larger systems such restrictions are not in force, and conical intersections are ubiquitous. The state energies themselves are sensitive to which root is chosen for optimization. Obviously, the chosen root has a better representation since the orbitals are optimized for it, while the non-chosen root has a poorer representation, and thus its energy is erroneously too high when computed as a root of the MCSCF reduced CI matrix. In situations where the two begin close to one another in energy, e.g., near a conical intersection, it is possible that ‘root switching’ will occur during the optimization process. Thus, in Figure 14.6, if one is at the geometrical position indicated by the asterisk and one selects the second root as the state for optimization (assuming the initial HF orbitals resemble better the orbitals of State A), as the MCSCF proceeds, the energy of the second root will drop below that of the first, since the orbitals are dropping the energy of State B at the expense of State A. As a result, the MCSCF will suddenly switch from optimizing the orbitals for State B to optimizing them for State A, as it is that state that is now the second root. This situation is unstable.

Figure 14.6 Two electronic states of an arbitrary system having a conical intersection. The inset region illustrates the effect on each curve of optimizing the orbitals for either State A or State B. At the coordinate position marked by an asterisk, the relative energies of the two states depend on which is chosen for orbital optimization, which can lead to root switching problems in an MCSCF calculation. Additionally, geometry optimization can cause root switching as well, if optimization passes through the conical intersection

To finesse this problem, it is possible to carry out a so-called ‘state-averaged’ MCSCF. In a state-averaged calculation, the orbitals are variationally optimized not for any one state energy, but rather for the average of the two (or more than two, if a larger number of states are of interest). A drawback to such a calculation is that the quality of any one state’s wave function is lower than it would be were it to be the only state under consideration. On the other hand, a virtue of a state-averaged calculation is that all states are expressed using the same MOs, thereby ensuring orthogonality, which is critical if, say, transition dipoles between states are to be computed.

Nevertheless, root switching may still be problematic for geometrical reasons in the vicinity of conical intersections. Thus, for instance, any optimization of State B in Figure 14.6 that begins to the left of the asterisk in coordinate q will ultimately proceed to the right until State B falls below State A in energy, at which point it is the first root for chemical reasons, not technical reasons. The only remedy in this situation is careful analysis in the construction of state PESs.

MCSCF results for phenylnitrene using a complete active space formed from the six phenyl π orbitals and the two nitrogen p orbitals and the eight electrons contained therein are presented in Table 14.1. Note that, because of symmetry and spin restrictions, only the 21A1 state must be determined as the second root of the MCSCF. The CAS results are quite

excited state and the ground state, there is a pole in the frequency-dependent polarizability, i.e., it diverges since the denominator goes to zero.

Using propagator methodology (sometimes also called a Green’s function approach or an equation-of-motion (EOM) method), the poles of the frequency-dependent polarizability can be determined without having to compute all of the necessary excited-state wave functions and their corresponding state energies. The necessary matrix equations are quite complex, and only a qualitative summary is provided here. Within the confines of the so-called randomphase approximation (RPA) the integrals that are required to compute the excitation energies are essentially those required to fill the CI matrix containing all single and double excitations and the transition dipole moments between the ground state and all singly excited configurations. Because the RPA method includes double excitations, it is usually more accurate than CIS for predicting excited-state energies. However, the method does not deliver a formal wave function, as CIS does. The RPA method may be applied to either HF or MCSCF wave functions. As with the CI formalisms they somewhat resemble, RPA solutions are most efficiently found by an iterative process that focuses only on a few lowest-energy excitations.

Table 14.2 includes RPA results for the six excited states of benzene already discussed in the context of CIS. The more complete RPA formalism does improve the results for those cases where CIS is most in error, but the net improvement in mean absolute error over all six states is only 0.1 eV in this case.

A DFT method that is strongly analogous to RPA is called time-dependent DFT (TDDFT). In this case, the KS orbital energies and various exchange integrals are used in place of matrix elements of the Hamiltonian. TDDFT is usually most successful for low-energy excitations, because the KS orbital energies for orbitals that are high up in the virtual manifold are typically quite poor. Casida, Casida, and Salahub (1998) have suggested that TDDFT results are most reliable if the following two criteria are met: (i) the excitation energy should be significantly smaller than the molecular ionization potential (note that excitations from occupied orbitals below the HOMO are allowed, so this is not a tautological condition) and (ii) promotion(s) should not take place into orbitals having positive KS eigenvalues.

Table 14.2 includes TDDFT results from the pure BPW91 and hybrid B3LYP functionals for the six excited states of benzene previously discussed for CIS and RPA methods. The pure functional is better for the most highly correlated 1B2u state, but the mean absolute error for the two methods over all six states is equivalent. The improved quality of the TDDFT results compared to CIS or RPA is substantial.

Many other comparisons between TDDFT and alternative methods have appeared. For example, Parac and Grimme (2002) found B3LYP TDDFT to give a mean unsigned error of 0.26 eV on the same 22 excited-state energies for 14 molecules discussed in the last section (cf. 0.14 eV for MR-MP2). In addition, Fabian (2001) compared B3LYP TDDFT results to CIS/INDO/S for various absorptions in 76 organosulfur compounds containing up to four sulfur atoms. The performance of TDDFT was again quite good: the mean unsigned error over all absorption maxima was 0.21 eV, which was superior to the CIS/INDO/S result of 0.35 eV. Interestingly the semiempirical PPP method, which by its nature is only applicable to excitations of the π → π * variety, had an error of only 0.20 eV, illustrating that more expensive calculations are not always better calculations . . .

A detailed comparison of several methods for local and charge-transfer excitation energies in benzenes substituted with donor and acceptor groups has been provided by Jamorski et al.

Table 14.3 Vertical excited-state energies (eV) of 4-dimethylaminobenzonitrile and 3,5-dimethyl- 4-dimethylaminobenzonitrile relative to S0 ground state

a Parusel (2000).

b Serrano-Andres´ et al. (1995).

c Parusel, Kohler,¨ and Grimme (1998).

d Jamorski et al. (2002); 6-311+G(2d,p) basis set. e Bulliard et al. (1999).

(2002) and some of their results are summarized in Table 14.3. An observation made in all of the above studies, and one that is particularly important for future developmental efforts, is that the TDDFT methodology performs relatively poorly for excitations characterized as charge-transfer (CT) or charge-resonance in weakly interacting composite chromophores (see also Casida et al. 2000 and Zyubin and Mebel 2003). Note, though, that the SCF approach works well when it is possible to employ it.

Efforts to improve TDDFT for higher-energy excitations have shown some early success. Tozer and Handy (1998) have proposed a correction procedure to deliver functionals having